A349274 Decimal expansion of the arc length of the logarithmic spiral related to a golden triangle with a unit base length.

4, 9, 2, 5, 1, 7, 5, 4, 3, 9, 4, 7, 8, 0, 5, 8, 8, 3, 7, 1, 9, 3, 1, 7, 6, 0, 4, 6, 8, 1, 3, 5, 6, 5, 6, 7, 4, 2, 2, 4, 5, 7, 0, 1, 9, 2, 8, 4, 0, 6, 9, 5, 2, 2, 2, 1, 5, 2, 0, 5, 8, 0, 2, 2, 0, 9, 4, 6, 3, 2, 5, 2, 4, 4, 3, 4, 6, 5, 2, 2, 6, 3, 3, 9, 0, 1, 5, 4, 3, 1, 5, 2, 0, 7, 8, 4, 4, 6, 8, 9, 5, 8, 3, 4, 8

Offset

1,1

Comments

The spiral starts at the triangle's apex (the vertex opposite to the base) and converges to the pole (the accumulation point).

References

H. E. Huntley, The Divine Proportion: A Study in Mathematical Beauty, New York: Dover Publications Inc., 1970, pp. 170-176.

Mario Livio, The Golden Ratio: The Story of Phi, The World's Most Astonishing Number, New York: Broadway Books, 2002, p. 119.

Links

Table of n, a(n) for n=1..105.

Amiram Eldar, Golden triangle and logarithmic spiral - illustration.

Hiroaki Kimpara, Logarithmic Spirals based on the Golden Ratio, Golden Square Ratio, Silver Ratio, and Silver Square Ratio, 2011.

Eric Weisstein's World of Mathematics, Golden Triangle.

Eric Weisstein's World of Mathematics, Logarithmic Spiral.

Wikipedia, Golden triangle (mathematics).

Wikipedia, Logarithmic spiral.

Formula

Equals sqrt((17 + 7*sqrt(5))/22) * sqrt(1 + b^2)/b, where b = 5*log(phi)/(3*Pi), and phi = (1+sqrt(5))/2 is the golden ratio (A001622).

Example

4.92517543947805883719317604681356567422457019284069...

Mathematica

b = 5 * Log[GoldenRatio]/(3*Pi); RealDigits[Sqrt[(17 + 7*Sqrt[5])/22] * Sqrt[1 + b^2]/b, 10, 100][[1]]

Crossrefs

Cf. A001622, A296794.

Sequence in context: A125575 A254159 A199725 * A021071 A197146 A021207

Adjacent sequences: A349271 A349272 A349273 * A349275 A349276 A349277

Keyword

nonn,cons

Author

Amiram Eldar, Nov 12 2021

Status

approved